Light propagation in curved spacetimes has long been a topic of great theoretical interest.1 Recently, the Event Horizon Telescope (EHT) observed the shadow of supermassive black holes from light propagating around the event horizon and has provided a way to understand black hole geometries.2–4 Nonlinear electrodynamics has also provided a nontrivial background for light propagation. Born and Infeld introduced a nonlinear Lagrangian for the Maxwell scalar and its pseudoscalar, which reduces to the Maxwell Lagrangian in the weak-field limit but leads to significant modifications of the physics of light–matter and light–light interactions in the strong-field limit.5 A detailed study has been made of the causal properties of the Born–Infeld Lagrangian.6 Heisenberg and Euler obtained an exact one-loop effective Lagrangian for electrons in a constant electromagnetic field.7 Later, Schwinger introduced the proper-time method for quantum field theory and obtained a one-loop effective Lagrangian for spinless charged bosons and spin-1/2 fermions in a constant electromagnetic field.8

The prominent features of the Heisenberg–Euler–Schwinger (HES) Lagrangian in quantum electrodynamics (QED) are vacuum polarization9,10 and Sauter–Schwinger pair production of electrons and positrons.8,11 Interestingly, in strong electric fields, the vacuum becomes unstable due to pair production, which is a consequence of the imaginary part of the HES Lagrangian.12,13 The polarized vacuum in a strong electromagnetic field provides a nontrivial background for propagation of probe photons (i.e., light). In recent years, nonlinear electrodynamics (NED) has been intensively studied, partly because of the development of ultra-intense lasers using chirped pulse amplification (CPA) technology14,15 and partly because of astrophysical observations of highly magnetized neutron stars and magnetars with magnetic fields comparable to or stronger than the critical field strength B_c = m²c²/eℏ = 4.4 × 10¹³ G.16 Such a strong electromagnetic field makes the vacuum into a polarized medium, and this polarized vacuum behaves similarly to a dielectric or ferromagnetic medium. NED in general exhibits, in addition to electric permittivity and magnetic permeability, the magneto-electric effect, in which a magnetic field induces electric polarization while an electric field induces magnetization.17–19 A strong electric field, on the other hand, creates pairs of charged particles and antiparticles through Sauter–Schwinger pair production.

NED thus exhibits a rich structure of vacuum polarization, involving phenomena such as vacuum birefringence, photon propagation, and the magneto-electric effect. In a strong magnetic field, the polarized vacuum acquires nontrivial refractive indices, which cause birefringence in the photon propagation either along the magnetic field or in the perpendicular plane.20,21 Light propagation has been intensively studied in the NED of Plebanski-type Lagrangians,22 according to which a low-energy probe photon satisfies a wave equation in a nonlinear background. Most studies in the literature have employed the covariant four-vector formulation of light propagation (see, e.g., Refs. 20 and 23–27), while the 3+1 formulation has been used for post-Maxwellian theory in weak electromagnetic fields.28

In a previous study, we investigated light propagation in a supercritical magnetic field and a weak electric field, in particular for the case of a nonzero electric field along the magnetic field (an “electromagnetic wrench”).29,30 In contrast to the pure magnetic field case, the electromagnetic wrench introduces magneto-electric effects and thus significantly modifies the light modes. In our previous work,29,30 we focused on a configuration with parallel electric and magnetic fields to study the effect in the simplest setting. However, given the complicated field configurations around neutron stars or in the focal regions of ultra-intense lasers, we need to extend the previous formulation to an arbitrary field configuration. Furthermore, it is worthwhile making the formulation applicable to other Lagrangians considered in the context of general NED.

In this paper, we introduce the 3+1 formulation of light modes in the NED of Plebanski-type Lagrangians L(F,G),22 where F and G are the Maxwell scalar and pseudoscalar, respectively:F≡14FμνFμν=12(B2−E2),G≡14FμνFμν*=−E⋅B.(1)Here, F = ∂A − ∂A is the field-strength tensor and F*μν=12εμναβFαβ (ɛ⁰¹²³ = 1) is the dual field-strength tensor.31 We also present a streamlined procedure for determining light modes with explicit formulas, which should be valuable for laboratory experiments and astrophysical observations of vacuum birefringence. As concrete examples, we apply the formulation to the frequently used NED Lagrangians for an arbitrarily strong magnetic field to obtain expressions for the light modes. These expressions can be used as optical tools to test theoretical black holes with NED Lagrangians. We use Lorentz–Heaviside units with c = ℏ = 1, and thus α_e = e²/4π, where e is the elementary charge.

In Minkowski spacetime with metric (+, −, −, −), the timelike Killing vector ∂_t makes the spacetime manifold foliate into one-parameter spacelike hypersurfaces. On each hypersurface Σ_t, the energy–momentum of a photon is given by k = (ω, k) and the field by F. In curved spacetime, the 3+1 formulation of Maxwell theory32–34 has often been preferred for modeling in astrophysics and cosmology. In the 3+1 formulation, the propagation and polarization of probe photons (light) are expressed in terms of directly measurable electromagnetic fields (E, B) and the photon direction k on each Σ_t. In practice, the 3+1 formulation has the advantage of providing the light modes in general NED in the familiar language of conventional optics, facilitating the design of experiments and observations.

The remainder of this paper is organized as follows. In Sec. II, we present the general 3+1 framework for analyzing light modes in a medium in the presence of strong background fields. Then, in Sec. III, this is compared with the formulation used in materials science to deal with the magneto-electric effect. In Sec. IV, we apply the framework to the NED of Plebanski-type Lagrangians and compare it with the alternative covariant formulation. In Sec. V, we present specific examples with the representative NED Lagrangians: the post-Maxwellian (PM), Born–Infeld (BI), ModMax (modified Maxwell, MM), and the HES QED Lagrangians. For these Lagrangians, the expressions for the light modes in an arbitrarily strong magnetic field are worked out. Finally, in Sec. VI, summarizing the results, we draw our conclusions, discussing the implications of the 3+1 formulation for research on vacuum birefringence, in the context, for example, of terrestrial experiments using ultra-intense lasers in the weak-field regime and of astrophysical observations of highly magnetized neutron stars in the strong-field regime.

In this section, we set up a 3+1 framework to obtain the light propagation modes in an arbitrary medium in the presence of strong background fields. When applied to NED Plebanski-type Lagrangians, as shown in Sec. IV, this framework yields refractive indices and polarization vectors for a given propagation direction.

In classical electrodynamics, the electromagnetic properties of a medium are described in terms of the polarization P, the magnetization M, the electric displacement D, and H:35D=E+P,H=B−M.(2)To find the modes of weak low-frequency36 probe light (δE, δB) in the medium polarized and magnetized by a strong slowly-varying background electromagnetic field (E₀, B₀), we decompose E, B, D, H, P, and M asA=A0+δA,(3)where A₀ and δA denote the background and probe parts, respectively.21,37 The background part A₀ is assumed uniform (locally constant) over the characteristic scales of the probe part δA, which allows us to argue that the relations (2) should be satisfied separately for the background and probe parts:D0=E0+P0,H0=B0−M0,δD=δE+δP,δH=δB−δM.(4)

Because the probe light is weak, only the linear response to the probe light is considered:δD=ϵE⋅δE+ϵB⋅δB,δH=μ̄B⋅δB+μ̄E⋅δE,(5)where ϵ_E, ϵ_B, μ̄B, and μ̄E (rank-2 tensors in the 3D space) are the permittivity (ϵ_E), permeability (μ̄B), and magneto-electric response (ϵ_B and μ̄E) tensors. These tensors, evaluated under condition (E, B) = (E₀, B₀), represent the influence of the strong background fields on the probe light. The tensors ϵ_B and μ̄E, absent in the theory of usual optical materials,38 represent the magneto-electric effect: the magnetic field induces polarization, and the electric field induces magnetization.39

When there are no real charge and current, the Maxwell equations for the probe light become−∂δB∂t=∇×δE,∂δD∂t=∇×δH,(6)which reduce toδB=n×δE,δD=−n×δH(7)for a Fourier mode characterized by an angular frequency ω and a propagation vector k=ωn=ωnn̂, where n is the refractive index of the mode, and δB, δE, δD, and δH vary as exp[iω(n · x − t)].

Then, substituting (7) into (5) leads to the master equation that determines the refractive indices and polarization vectors n and δE for a given propagation direction n̂:ϵE⋅δE+ϵB⋅n×δE+n×μ̄B⋅n×δE+μ̄E⋅δE=0.(8)This equation can be written in matrix–vector form asΛ⋅δE=ϵE+ñ⋅μ̄E+ϵB⋅ñ+ñ⋅μ̄B⋅ñ⋅δE=0,(9)where ñij=−ϵijknk.40 Solving the equation involves two steps: solving det(Λ(n)) = 0 to find a value of n and then solving Λ(n) · δE(n) = 0 to find δE(n) associated with the value. Usually, multiple values of n are obtained, implying multirefringence. The relation between the polarization vectors associated with different values of n can hardly be known before finding the vectors explicitly, because they refer to different Λ matrices. But the relation between the polarization vectors of a given value of n may be known from the symmetry of Λ. To solve the equation, we need the explicit forms of ϵ_E, ϵ_B, μ̄B, and μ̄E, which are derived for Plebanski-type Lagrangians in Sec. IV A. From now on, δE refers to the probe electric field without the factor exp[iω(n · x − t)].

The 3+1 framework in Sec. II closely follows the approach in conventional optics and thus can facilitate direct comparison of the magneto-electric effects in field-modified vacua with those in optical media. Although magneto-electric effects are rarely observed in the usual optical media, some multiferroic materials have been known to exhibit these effects.18,19,41 In this section, we compare the 3+1 framework with that used in materials science, in which the material response is found by minimizing the Helmholtz free energy F:18F(E,H)=−12E⋅ϵ⋅E−12H⋅μ⋅H−E⋅α⋅H,P=−∂F∂E=ϵ⋅E+α⋅H,M=−∂F∂H=μ⋅H+αT⋅E,(10)where ϵ and μ (symmetric) are the permittivity and permeability tensors, respectively. The magneto-electric response tensor α is odd under time reversal or space inversion to keep F(E,H) invariant under such transformations.35 Here, all the quantities are real.

To compare (10) with (5), we express δP and δM in (4) and (5) in terms of δE and δH. Substituting δB=δH+δM into (5) while assuming that μ̄B−1 exists, we obtainδP=ϵE−1−ϵB⋅μ̄B−1⋅μ̄E⋅δE+ϵB⋅μ̄B−1⋅δH,δM=μ̄B−1−1⋅δH−μ̄B−1⋅μ̄E⋅δE.(11)For the magneto-electric parts in (11) to match those in (10), ϵ, μ, and α should be related to ϵ_E, ϵ_B, μ̄B, and μ̄E as follows:ϵ=ϵE−1−ϵB⋅μ̄B−1⋅μ̄E,μ=μ̄B−1−1,α=ϵB⋅μ̄B−1=−μ̄B−1⋅μ̄ET.(12)These relations and the symmetry of ϵ and μ lead to the relationsϵE=ϵET,μ̄B=μ̄BT,ϵB=−μ̄ET,(13)which also hold for Plebanski-type Lagrangians, as will be shown in Sec. IV.

The relation reciprocal to (12) and (13) is obtained by expressing P and M in (10) in terms of E and B, which establishes the equivalence between the linear response coefficients in the materials science literature and those in our 3+1 formulation:ϵE=ϵ−α⋅(1+μ)−1⋅αT+1,ϵB=α⋅(1+μ)−1,μ̄B=(1+μ)−1,μ̄E=−(1+μ)−1⋅αT.(14)By substituting these expressions into (9) and rearranging terms, we obtain a factorized form of Λ:Λ=1+ϵ−ñ+α⋅(1+μ)−1⋅ñ+αT,(15)from which the property Λ = Λ is obvious. Because Λ is real-symmetric, the polarization vectors associated with a value of n can be made orthonormal to one another.42 The factorization appears to be useful, but the specific forms of ϵ, μ, and α would be necessary for further analysis.

In this section, we derive the general expressions for the light propagation modes in NED described by Plebanski-type Lagrangians. The general framework of mode analysis in Sec. II is used to find the refractive indices and polarization vectors for a given propagation direction.

When a strong electromagnetic field modifies the vacuum, the corresponding Lagrangian should be an analytic function of the Maxwell scalar F and Maxwell pseudoscalar G to satisfy Lorentz and gauge invariance. Such Lagrangians are called Plebanski-type Lagrangians:22L=L(F,G)=−F+L′,(16)where L′ represents the non-Maxwell part. Typically, L(F,G) is an even function of G to preserve parity.

In general, the Lagrangian may have an imaginary part, which leads to absorption of probe light.43 For example, the HES Lagrangian has an imaginary part when either only an electric field exists or the electric field has a component parallel to the magnetic field. This imaginary part becomes significant as the purely electric field or the electric field component along the magnetic field approaches or surpasses the critical field strength Ec=me2/e=1.32×1016V/cm2.44 As a consequence, electron–positron pairs are produced by strong electromagnetic fields, in Sauter–Schwinger pair production or vacuum breakdown.8,11 In the present work, we do not take such an imaginary part into account, to assume a stable medium. In practice, the imaginary term can be neglected when the electric field is weaker than E_c/3 and the magnetic field.30

Once the Lagrangian is given, the polarization P and magnetization M are obtained by differentiating the Lagrangian with respect to the electric and magnetic fields:7,45D=E+P=∂L∂E,H=B−M=−∂L∂B.(17)For Plebanski-type Lagrangians, we obtainD=−LFE−LGB,H=−LFB+LGE,(18)where the subscripts F and G denote partial differentiation of L with respect to the respective quantities: for example, LFG≡∂2L/(∂F∂G).

Then, we can obtain the permittivity, permeability, and magneto-electric response tensors by taking a variation of E and B in (18) and applying the decomposition argument in Sec. II:ϵE=−LFI+[LFFE0E0+LFG(E0B0+B0E0)+LGGB0B0],ϵB=−LGI+[−LFFE0B0+LFG(E0E0−B0B0)+LGGB0E0],μ̄B=−LFI+[−LFFB0B0+LFG(B0E0+E0B0)−LGGE0E0],μ̄E=LGI+[LFFB0E0+LFG(B0B0−E0E0)−LGGE0B0],(19)where the products of two vectors denote dyadics. Here, L_F, L_G, L_FF, L_FG, and L_GG denote LF, LG, LFF, LFG, and LGG evaluated under condition (E, B) = (E₀, B₀). From now on, we omit the subscript 0 so as not to clutter the notation. With these expressions for the linear response tensors, we can solve the master Eq. (9) to find the light modes, as will be shown in Sec. IV B. A similar formulation was presented by Robertson:46 the modes were analyzed for weakly nonlinear Lagrangians and a fixed propagation direction, and the coupling to axions was considered.

The linear response tensors in (19) exhibit some symmetries. First, ϵE=ϵET, μ̄B=μ̄BT, and ϵB=−μ̄ET, i.e., (13). Second, ϵE↔μ̄B and ϵB↔−μ̄E under the duality transformation (E, B) → (−iB, iE), which keeps F, G, and L the same. Finally, ϵ_E and μ̄B have even parity, while ϵ_B and μ̄E have odd parity: L should be an even function of G, i.e., L(F,−G)=L(F,G), to respect inversion symmetry.

In this subsection, we systematically solve the master Eq. (9) with the tensors (19) to find the light modes for an arbitrary propagation direction. The corresponding Λ in (9) is now given byΛ=−LF[(1−n2)I+nn]+LFFPP+LFG(PQ+QP)+LGGQQ,(20)whereP=E+n×B,Q=B−n×E.(21)From this dyadic expression for Λ, it is clear that Λ is real-symmetric as in the case of materials science, which is a feature of Plebanski-type Lagrangians. Through attempting to find complete squares of P and Q in Λ, we express Λ as a linear combination of the identity matrix and self-conjugate dyadics, i.e., dyadics of two identical vectors:47Λ=XX+YY+ZZ+U,(22)whereX=aP+bQ,Y=cQ,Z=dn,U=eI,(23)anda2=LFF,ab=LFG,b2=LFG2LFF,c2=LFFLGG−LFG2LFF,d2=−LF,e=−LF(1−n2).(24)

The form of Λ in (22) implies that the reciprocal vectors of X, Y, and Z may be used as basis vectors to represent δE, as in crystallography. However, this requires nondegeneracy, i.e., the condition that the volume V ≔ X × Y · Z is not zero. The degeneracy (V = 0) is caused by either acd = 0 or P×Q⋅n̂=0, as can be found from (21) and (23). The first condition acd = 0 is independent of n̂ and depends only on the functional form of L(F,G) and the background field. In fact, the condition reduces to c = 0, because a² = L_FF and d² = −L_F do not vanish for nontrivial Lagrangians. An example of having c = 0 is the MM Lagrangian, as will be shown in Sec. V C. By contrast, the second condition P×Q⋅n̂=0 depends also on n̂. The second condition is met by a configuration satisfying Bt=n̂×Et,48 which we call the free-propagation configuration (FPC). In the FPC, n is unity, and S ≔ E × B heads along n̂, and any vector perpendicular to n̂ is a polarization vector. In other words, the light in FPC propagates as if it were in a free vacuum. In the nondegenerate case, the FPC is the only configuration with n = 1. In the degenerate case of c = 0, however, another configuration has n = 1, as shown in Sec. IV B 2. A special case of FPC is the collinear configuration: E‖B‖n̂. Below, we deal with the nondegenerate case of c ≠ 0 first and then the degenerate case of c = 0, disregarding the FPC, owing to its triviality.

Nondegeneracy (V ≠ 0) allows us to represent δE as a linear combination of the reciprocal vectors of X, Y, and Z:δE=pX′+qY′+rZ′,(25)whereX′=Y×ZV,Y′=Z×XV,Z′=X×YV.(26)The linear equations for p, q, and r are then obtained by multiplying (inner product) each of X, Y, and Z from the left into the master equation Λ · δE = 0 with Λ given by (22):Mp+Hq=0,Hp+Nq=0,r=−Z⋅Xd2p−Z⋅Yd2q,(27)whereH=X⋅Y−(Z⋅X)(Z⋅Y)d2,M=X2+e−(Z⋅X)2d2,N=Y2+e−(Z⋅Y)2d2.(28)It is assumed that d² = −L_F ≠ 0, which holds in general because any nonlinear Lagrangian should have the Maxwell term, i.e., −F, as the leading term.

The linear equations in p, q, and r(27) are solved in two steps. In the first step, the null determinant condition, MN − H² = 0, is solved for n. In the second step, each n is substituted into (27), which is then solved for (p, q, r), shown in Table I. In the particular case of H = M = N = 0, (p, q) can have arbitrary values, and the resulting polarization vector spans a plane:δE=pX′−X⋅Zd2Z′+qY′−Y⋅Zd2Z′.(29)

In contrast to the photon in a free vacuum, a photon in the NED of a Plebanski-type Lagrangian acquires a longitudinal component when r ≠ 0: δE · n = r/d from (23), (25), and (26). Such a longitudinal component appears in anisotropic dielectric media.38

Now we solve MN − H² = 0 for the refractive indices. Factorizing MN − H² by using (21), (23), (24), and (28), we obtain the Fresnel equation:P2̃n2−1−λ(+)P2̃n2−1−λ(−)=0,(30)whereP2̃:=P2−n2Pn2=E2−2nSn+n2Bt2−En2,λ(±)=−β±β2−4γα2γ.(31)The parameters α, β, and γ are determined solely by the background field, regardless of n̂, and thus so is λ⁽⁾:α=LF2+2LF(GLFG−FLGG)−LFFLGG−LFG2G2=d4−2d2[abG−(b2+c2)F]−a2c2G2,β=LF(LFF+LGG)−2FLFFLGG−LFG2=−d2(a2+b2+c2)−2a2c2F,γ=LFFLGG−LFG2=a2c2.(32)Equation (30) can be rewritten asBt2−En2−λ(±)n2−2Snn+En2+λ(±)=0,(33)from which we obtain the refractive indices:n±(i)=−Sn±Sn2+λ(i)−Bt2+En2λ(i)+E2λ(i)−Bt2+En2,(34)where i = ±. Without loss of generality, we take only positive values of n and solve (27) to obtain the corresponding polarization vectors (see Table I). It should be mentioned that we may have multirefringence, even beyond birefringence, because (30) is a quartic equation in n.49 For a purely magnetic background field, (n(i))2 has the simpler formn(i)2=λ(i)λ(i)−Bt2,(35)implying birefringence.

So far, we have implicitly assumed that n ≠ 1. When n = 1, the dyadic Λ in (22) becomes Λ = XX + YY + ZZ because of e = 0. Then, the condition MN − H² = 0 reduces to P_t = Q_t = 0, which is satisfied only for the FPC. In such a case, the light sees no effect of NED.

The condition c = 0 nullifies Y in (23) and thus simplifies Λ. Unless Z‖X, which turns out to be the FPC, we can proceed similarly to the nondegenerate case by introducing two-dimensional reciprocal vectors:X′=Z2X−(X⋅Z)ZZ2X2−(X⋅Z)2,Z′=X2Z−(Z⋅X)XX2Z2−(Z⋅X)2,(36)where X′ and Z′ span the same plane spanned by X and Z; δE has no component perpendicular to the plane. The counterparts to (25), (27), and (28) become as follows:δE=pX′+rZ′,M̃p+H̃r=0,H̃p+Ñr=0,H̃=X⋅Z,Ñ=Z2+e,M̃=X2+e.(37)The null determinant condition M̃Ñ−H̃2=0 is given as a quadratic equation in n:a2En2−Bt2+b2Bn2−Et2−LF−2abGn2+2Sn(a2+b2)n−a2E2+2abG−b2B2+LF=0.(38)Once n is available from this equation, the polarization vectors can be found from (p, r) in Table I.

In contrast to the nondegenerate case, the degenerate case always has a non-FPC with n = 1. When n = 1, Λ = XX + ZZ. When Z ∦ X (non-FPC), Z × X becomes the polarization vector, because Λ · δE = 0 is satisfied by construction. The case Z‖X corresponds to the FPC. An advantage of our 3+1 formulation is a transparent analysis of the degenerate case compared with that of the covariant formulation.

The procedure to find light modes in nondegenerate nonlinear electrodynamics is outlined in Table II. Here, we succinctly present the essential formulas used for the procedure in order. It is assumed that the formulas for {L_F, L_FF, L_FG, L_GG}, and thus {a², ab, b², c², d²} in (24) are known, regardless of the field configuration.

Given a configuration specified by {E,B,n̂} in the 3+1 formulation, we then specialize {a², ab, b², c², d²} to the configuration and calculate {α, β, γ} in (32) to obtain the refractive index n from (34). The refractive index may need to be numerically evaluated if the Lagrangian has a complicated functional form. With the refractive index, we calculate P and Q in (21) and the following auxiliary parameters:Pc=Ec+nBt,Qc=Bc−nEt,P×Q=S−n(E2+B2)n̂+n(EnE+BnB)+n2Snn̂,PQ̃≔P⋅Q−n2PnQn=−(1−n2)G,P2̃≔P2−n2Pn2=E2−2nSn+n2Bt2−En2,Q2̃≔Q2−n2Qn2=B2−2nSn+n2Et2−Bn2,(39)where the subscripts n and t denote components respectively parallel and transverse to n.

Then, acH, M, and N in (28) are expressed asacH=a2c2PQ̃+abc2Q2̃,M=a2P2̃+2abPQ̃+b2Q2̃+d2(1−n2),N=c2Q2̃+d2(1−n2).(40)Unless acH = M = N = 0, we form the following two vectors proportional to the polarization vectors in Table I:δE1=a2MPc+(acH+abM)Qc−(a2Pn+abQn)acH−a2c2MQnd2P×Q,δE2=−acHa2c2Pc−a2N+acHabc2Qc+a2[(a2Pn+abQn)N−acHQn]d2P×Q.(41)We can choose any nonzero vector from δE₁ and δE₂ as the polarization vector for the refractive index n. When both are nonzero, they differ from each other only by a multiplicative factor. When acH = M = N = 0, the polarization vector spans a plane:δE=[d2Qc−(a2Pn+abQn)P×Q]p+(a2d2Pc+abd2Qc+a2c2QnP×Q)q,(42)where p and q are arbitrary real numbers.

In summary, light modes are completely determined by the formulas (32), (34), and (40)–(42) once {a², ab, b², c², d²} (24) and the configuration parameters (39) are known.

The most economical way to obtain light modes is the covariant four-vector formulation, to which our 3+1 formulation is equivalent in this regard. With the photon propagation four-vector k = ω(1, n), the covariant vectors used for polarization tensors take the formFμνkν=ω(E⋅n,P),F*μνkν=ω(B⋅n,Q).(43)The Fresnel equation in the covariant formulation, for instance Eq. (12) for f2≔FαμFαkμνkν in De Lorenci et al.49 recovers (30) with the same {α, β, γ}.

However, the 3+1 formulation has a more direct connection to the usual concepts in optics, such as the permittivity, permeability, and magneto-electric response tensors (19), and provides the refractive indices and polarization vectors in terms of familiar quantities E, B, and n̂. Thus, it can allow a richer comparison of the nonlinear vacua with conventional optical media. The formulation also reveals the unique mathematical structure of the nonlinear vacuum’s response: the Fresnel matrix Λ consists of self-conjugate dyadics (22), which facilitates the construction of polarization vectors. Finally, the formulation is related to the 3+1 formulation in curved spacetime,32 which is popular in numerical simulation of general relativistic phenomena.50 In our 3+1 formulation in Minkowski spacetime, we use the Killing vector ∂_t to decompose the spacetime into one-parameter spacelike hypersurfaces Σ_t. Each observer on Σ_t measures the energy–momentum of photons as k and the electromagnetic field E, B.

In this section, we apply the 3+1 formulation to several NED Lagrangians for the case of a purely magnetic background field: post-Maxwellian (PM),28 Born–Infeld (BI),5 ModMax (MM),51,52 and Heisenberg–Euler–Schwinger (HES) QED53 Lagrangians. The PM Lagrangians are frequently used in X-ray polarimetry for light propagation near pulsars and also in black hole physics. Recently, these NED Lagrangians and Plebanski-type Lagrangians, in general, have been actively used for theoretical black holes.54 Theoretical magnetic black holes whose magnetic field strength is at electroweak scales have been proposed,55 and these require QED corrections in the context of general relativity.

Without loss of generality, we assume that the magnetic field is along the z axis, and the light’s propagation direction vector lies on the xz plane: B=B0ẑ and n̂=(sin⁡θ,0,cos⁡θ), as shown in Fig. 1. The light modes for a given Lagrangian, i.e., n and δE, are found by following the procedure in Table II and Sec. IV C, as summarized in Table III.

Interestingly, the light propagation modes of the four Lagrangians exhibit a common structure, as manifestly parameterized in Table III, owing to the fact that these Lagrangians are of Plebanski type. The polarization vector is either perpendicular to the plane (⊥) formed by the magnetic field and the propagation direction or on the plane (‖). As a consequence of the common structure, the photon propagation at special angles is universal. When θ = 0, n_⊥ = n_‖ = 1 with δE_‖ = (1, 0, 0): a photon propagating along the magnetic field sees no background field effect. Note that δE_⊥ = (0, 1, 0) regardless of θ. When θ = π/2, δE_‖ = (0, 0, 1): a photon propagating perpendicular to the magnetic field has no longitudinal component. In general, nonzero ϵ implies the presence of a longitudinal component: δE‖⋅n̂=(sin⁡θ⁡cos⁡θ)ϵ/(1+ϵ). The details specific to each Lagrangian are discussed below.

The PM Lagrangian28 is given byLPM=−F+η1F2+η2G2,(44)where η₁ and η₂ are parameters. It is useful as the generic weak-field form of the Plebanski Lagrangians that approaches the Maxwell Lagrangian (−F) as the field vanishes. For example, η₁ = η₂ = 1/(2T) and η₁/4 = η₂/7 = e⁴/(360π²m⁴) give the BI and HES7,8,56 Lagrangians in the weak-field limit, respectively. The PM Lagrangian has been widely used to design vacuum birefringence experiments, since only subcritical fields are available in the laboratory, even with the most intense lasers.44,57,58 One may extend LPM by adding a product of F and G: a parity-violating term η₃FG or a parity-conserving term η3FG2.28

The procedure in Table II yields the coefficients (24) and (32), regardless of the specific configuration:a2=2η1,ab=0,b2=0,c2=2η2,d2=1−2Fη1,α=4F2η12−4(2F2+G2)η1η2−4F(η1−η2)+1,β=4Fη12−4Fη1η2−2(η1+η2),γ=4η1η2.(45)

The light modes in the presence of a purely magnetic background field are given in Table III, and they reduce in the weak-field limit to the well-known modes n⊥=1+η1B02⁡sin2⁡θ, with δE_⊥ = (0, 1, 0), and n‖=1+η2B02⁡sin2⁡θ, with δE_‖ = (cos θ, 0, − sin θ).21 The parallel mode has a longitudinal component: δE‖⋅n̂=η2B02⁡sin⁡2⁡θ.

The BI Lagrangian, introduced to remove the self-energy divergence in classical electrodynamics, is given by5LBI=T−T2+2FT−G2,(46)where T is a parameter. In the weak-field limit (T ≫ F, G), the BI Lagrangian becomes the PM Lagrangian with η₁ = η₂ = 1/(2T).

The coefficients (24), (31), and (32) are given bya2=T2D3/2,ab=−GTD3/2,b2=G2D3/2,c2=1D1/2,d2=TD1/2,α=T2(2F+T)2D2,β=−2T2(2F+T)D2,γ=T2D2,λ(±)=2F+T,(47)where D=2FT−G2.

As shown in Table III, the BI Lagrangian does not have birefringence in a pure magnetic field.52 Furthermore, H = M = N = 0 holds (see Table I), and thus the polarization vector spans a plane as in a free vacuum, but possibly with a nonzero longitudinal component: δE⋅n̂=qδE‖⋅n̂=qB02⁡sin⁡2⁡θ/(2T).

The MM Lagrangian, introduced as a modification of the Maxwell theory to preserve duality invariance and conformal invariance,51 is given byLMM=−F⁡cosh⁡g+F2+G2sinh⁡g,(48)where g ≥ 0 for a lightlike or subluminal propagation of light. It reduces to the Maxwell Lagrangian when g = 0.

The coefficients (24) are found asa2=G2R3sinh⁡g,ab=−FGR3sinh⁡g,b2=F2R3sinh⁡g,c2=0,d2=cosh⁡g−FRsinh⁡g,(49)where R=F2+G2. As c² = 0 (a consequence of the conformal and duality invariance), the MM Lagrangian is degenerate, and the formulas in Sec. IV B 2 are used for analysis.

The light modes in a pure magnetic field are shown in Table III. In contrast to the modes for other Lagrangians, the refractive indices and polarization vectors are independent of the strength of the background magnetic field, which is a consequence of conformal invariance.52 The perpendicular mode always has n_⊥ = 1, regardless of the configuration, as discussed in Sec. IV B 2. A photon in this mode propagates as if it sees no background field.

Heisenberg, Euler, and Schwinger obtained the one-loop correction to the Maxwell Lagrangian due to spin-1/2 fermions of mass m in constant electric and magnetic fields of arbitrary strengths.7,8 The exact one-loop action is given as a proper-time integral:L(1)(h,g)=−18π2∫0∞dse−m2ss3ehs⁡coth(ehs)egs⁡cot(egs)−1+(ehs)2−(egs)23,(50)whereh=F2+G2+F,g=F2+G2−F.(51)The HES Lagrangian is obtained by adding the correction to the Maxwell Lagrangian:LHES(h,g)=g2−h22+L(1)(h,g).(52)

The direct analytical form of LHES for arbitrary constant electromagnetic fields is not yet known. However, the direct form for the case of G = 0 was obtained using dimensional regularization,53 and it was confirmed by gamma-function regularization in the in–out formalism:59LHESG=0(h̄)=m48π2h̄2−π4αe+ζ′(−1,h̄)+m48π2h̄2−112+h̄24−112−h̄2+h̄22lnh̄,(53)where h̄=m2/(2eh), α_e = e²/(4π) is the fine structure constant, ζ(s,h̄) is the Hurwitz zeta function, and ζ′(−1,h̄)=dζ(s,h̄)/ds|s=−1. For the case with G ≠ 0, the correction was obtained up to the order of G⁴.60 Recently, we have expressed the correction as a power series in a small parameter (g/h).30

In finding the light modes in a pure magnetic field, one may be tempted to use the Lagrangian for a pure magnetic field, LHESG=0(h̄)|h=B0. However, the Lagrangian is not sufficient, and a correction of O(G) [equivalently O(g/h)] should be included because the combined fields of the background field and the light field can have G ≠ 0 in general (δE · B₀ is not necessarily zero). The correction is given byLcor(h̄,ḡ)=m48π2h̄21ḡ2−124h̄+lnh̄−ψ(0)12,(54)where ḡ=m2/(2eg), and ψ(0)(h̄) is the zeroth-order polygamma function.30 The coefficients {a², ab, b², c², d²} are then obtained from (24) by applying the conditions h = B₀ and g = 0 (a purely magnetic background field):a2=16αe2h̄2m42h̄2(ψ(0)(h̄)−1)+h̄1−lnh̄Γ(h̄)22π+13,ab=b2=0,c2=−8αe2h̄23m46h̄2−6h̄ln2πh̄Γ(h̄)2−8αe2h̄23m41+2ψ(0)(h̄)−24ζ′(−1,h̄)+1h̄,d2=αe6π6h̄2−6h̄ln2πh̄Γ(h̄)2−24ζ′(−1,h̄)+2⁡lnh̄+1+1.(55)With {a², ab, b², c², d²} and {B0,n̂}, we find the light modes in the QED vacuum in an arbitrarily strong magnetic field, as shown in Table III, by following the procedure in Table II and Sec. IV C. The simple expressions for the modes in Table III were obtained by Melrose61 and Denisov et al.,62 although they gave the coefficients μ and ϵ as complicated integrals instead of explicit functions as in (55).